The λ – Calculus

... said to be bound in the sub-expression M 2. A bound variable is one whose name is the same as the parameter. Otherwise the variable is said to be free 3. Any variable not bound in M is said to be free Note: bound variables are placeholders just like function parameters in the imperative and OOP para ...

... said to be bound in the sub-expression M 2. A bound variable is one whose name is the same as the parameter. Otherwise the variable is said to be free 3. Any variable not bound in M is said to be free Note: bound variables are placeholders just like function parameters in the imperative and OOP para ...

LambdaCalculus

... represents the same f function, except it is anonymous. To represent the function evaluation f(2) = 4, we use the following -calculus syntax: ...

... represents the same f function, except it is anonymous. To represent the function evaluation f(2) = 4, we use the following -calculus syntax: ...

Lambda Calculus

... represents the same f function, except it is anonymous. To represent the function evaluation f(2) = 4, we use the following λ-calculus syntax: (λx. x2 2) ⇒ 22 ⇒ 4 ...

... represents the same f function, except it is anonymous. To represent the function evaluation f(2) = 4, we use the following λ-calculus syntax: (λx. x2 2) ⇒ 22 ⇒ 4 ...

Lecture 10

... The 3 argument version of the reduce function • We have already seem reduce(function, list) • There is a 3 argument variant which is • Reduce(function, list, identity element/first element for the process ...

... The 3 argument version of the reduce function • We have already seem reduce(function, list) • There is a 3 argument variant which is • Reduce(function, list, identity element/first element for the process ...

Functional Programming Languages

... set, called the domain set, to another set, called the range set • A lambda expression specifies the parameter(s) and the mapping of a function in the following form (x) x * x * x for the function cube(x) = x * x * x ...

... set, called the domain set, to another set, called the range set • A lambda expression specifies the parameter(s) and the mapping of a function in the following form (x) x * x * x for the function cube(x) = x * x * x ...

Lambda Calculus and Lisp

... • Lisp was the first FL and is the one most people think of. It has a simple syntax using prefix notation and parentheses. • Scheme is a dialect of Lisp. It has static scope rather than dynamic, uses meaningful identifiers, true and false are #T and #F, predicates end in ? ( so (atom? (x) ) returns ...

... • Lisp was the first FL and is the one most people think of. It has a simple syntax using prefix notation and parentheses. • Scheme is a dialect of Lisp. It has static scope rather than dynamic, uses meaningful identifiers, true and false are #T and #F, predicates end in ? ( so (atom? (x) ) returns ...

Lecture 15: The Lambda Calculus

... • Homework: Assignment 10 in the workbook • Thu 3 Dec: tutorial on Assignment 10, general revision • Fri 4 Dec: Class Test 2, 10am, Assembly Hall ...

... • Homework: Assignment 10 in the workbook • Thu 3 Dec: tutorial on Assignment 10, general revision • Fri 4 Dec: Class Test 2, 10am, Assembly Hall ...

Notes

... • Variable x does not evaluate to anything and immediately results in an error. For this to happen, x must have been free in the expression that is being evaluated. Since it was free, there is no value that can be associated with it. 1 These are the call-by-value semantics for the lambda calculus. O ...

... • Variable x does not evaluate to anything and immediately results in an error. For this to happen, x must have been free in the expression that is being evaluated. Since it was free, there is no value that can be associated with it. 1 These are the call-by-value semantics for the lambda calculus. O ...

C311 First Class Objects

... Expressible as an anonymous literal value Storable in variables Storable in data structures Having an intrinsic identity (independent of any given name) Comparable for equality with other entities Passable as a parameter to a function Returnable as the result of a function call Constructable at runt ...

... Expressible as an anonymous literal value Storable in variables Storable in data structures Having an intrinsic identity (independent of any given name) Comparable for equality with other entities Passable as a parameter to a function Returnable as the result of a function call Constructable at runt ...

function

... 1. If the free variables of N have no bound occurrences in M, then the term M[N/x] is formed by replacing all free occurrences of x in M by N. 2. Otherwise, assume that the variable y is free in N and bound in M. Consistently replace the binding and the corresponding bound occurrences of y in M by a ...

... 1. If the free variables of N have no bound occurrences in M, then the term M[N/x] is formed by replacing all free occurrences of x in M by N. 2. Otherwise, assume that the variable y is free in N and bound in M. Consistently replace the binding and the corresponding bound occurrences of y in M by a ...

PPT

... (function (f) { return function (x) { return f(f(x)); } ; ) (function (y) { return y +1; }) function (x) { return (function (y) { return y +1; }) ((function (y) { return y + 1; }) (x)); } function (x) { return (function (y) { return y +1; }) (x + ...

... (function (f) { return function (x) { return f(f(x)); } ; ) (function (y) { return y +1; }) function (x) { return (function (y) { return y +1; }) ((function (y) { return y + 1; }) (x)); } function (x) { return (function (y) { return y +1; }) (x + ...

Functional Programming

... parameters and yields a function whose value is the first actual parameter function applied to the application of the second Form: h f ° g (or g ; f) which means h (x) f (g ( x)) For f (x) x * x * x and g (x) x + 3, h f ° g yields (x + 3)* (x + 3)* (x + 3) ...

... parameters and yields a function whose value is the first actual parameter function applied to the application of the second Form: h f ° g (or g ; f) which means h (x) f (g ( x)) For f (x) x * x * x and g (x) x + 3, h f ° g yields (x + 3)* (x + 3)* (x + 3) ...

Functional Programming

... Referential Transparency • Each expression can be replaced with its value without changing the behavior of the program. • There is no distinction between a reference to a thing and the thing itself (e.g., call-by-value and call-by-reference are the same). • The order in which expressions are evalua ...

... Referential Transparency • Each expression can be replaced with its value without changing the behavior of the program. • There is no distinction between a reference to a thing and the thing itself (e.g., call-by-value and call-by-reference are the same). • The order in which expressions are evalua ...

Breck, Hartline

... the toolbox to IMP. IMP has no functions, (procedures, routines, methods). It has no way to write a functional abstraction. This makes it pretty uninteresting as a programming language. So we will overcompensate and introduce a language with nothing but functional abstraction: the lambda calculus. T ...

... the toolbox to IMP. IMP has no functions, (procedures, routines, methods). It has no way to write a functional abstraction. This makes it pretty uninteresting as a programming language. So we will overcompensate and introduce a language with nothing but functional abstraction: the lambda calculus. T ...

Functional Programming

... It is ambiguous : we do not know which names are parameters and which ones are global “lambda expressions” come in hand (mapcar ‘(lambda (x) (cons val x)) L) ...

... It is ambiguous : we do not know which names are parameters and which ones are global “lambda expressions” come in hand (mapcar ‘(lambda (x) (cons val x)) L) ...

Lecture10 - CIS @ UPenn

... The 3 argument version of the reduce function • We have already seem reduce(function, list) • There is a 3 argument variant which is • Reduce(function, list, identity element/first element for the process • Sorting examples • Define insertion of an element into a sorted list • Now use a reduce oper ...

... The 3 argument version of the reduce function • We have already seem reduce(function, list) • There is a 3 argument variant which is • Reduce(function, list, identity element/first element for the process • Sorting examples • Define insertion of an element into a sorted list • Now use a reduce oper ...

Functional Programming COMP2003

... situation, and so can be defined and used in just one place. Usually the motive for forming the computation into a function is that it is being passed in as argument to some other functions which will use that computation in various ways, probably applying it repeatedly 2. When there is a need to ma ...

... situation, and so can be defined and used in just one place. Usually the motive for forming the computation into a function is that it is being passed in as argument to some other functions which will use that computation in various ways, probably applying it repeatedly 2. When there is a need to ma ...

15. Functional Programming

... called the domain set, to another set, called the range set A lambda expression specifies the parameter(s) and the mapping of a function in the following form λ(x) x * x * x for the function cube (x) = x * x * x ...

... called the domain set, to another set, called the range set A lambda expression specifies the parameter(s) and the mapping of a function in the following form λ(x) x * x * x for the function cube (x) = x * x * x ...

Lecture10

... The 3 argument version of the reduce function • We have already seem reduce(function, list) • There is a 3 argument variant which is • Reduce(function, list, identity element/first element for the process ...

... The 3 argument version of the reduce function • We have already seem reduce(function, list) • There is a 3 argument variant which is • Reduce(function, list, identity element/first element for the process ...

ppt

... alpha-renaming, beta reduction, eta conversion, applicative and normal evaluation orders, ChurchRosser theorem, combinators, booleans ...

... alpha-renaming, beta reduction, eta conversion, applicative and normal evaluation orders, ChurchRosser theorem, combinators, booleans ...

Lambda calculus

... • The lambda calculus was introduced by mathematician Alonzo Church in the 1930s . • The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox. • Subsequently, in 1936 Church isolated and published just the portion rel ...

... • The lambda calculus was introduced by mathematician Alonzo Church in the 1930s . • The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox. • Subsequently, in 1936 Church isolated and published just the portion rel ...

Introduction to Lambda Calculus - CSE IITK

... add x y = x + y inc = add 1 map f [] = [] map f (x:xs) = f x : map f xs • map is a higher order function. It takes a function as argument. • Functional programming treats functions as firstclass citizens. There is no discrimination between function and data. map inc [1, 2, 3] => ...

... add x y = x + y inc = add 1 map f [] = [] map f (x:xs) = f x : map f xs • map is a higher order function. It takes a function as argument. • Functional programming treats functions as firstclass citizens. There is no discrimination between function and data. map inc [1, 2, 3] => ...

functional form

... • Lambda expressions are earlier methods for describing nameless functions • Lambda expressions are applied to parameter(s) by placing the parameter(s) after the expression e.g., ((x) x * x * x)(2) which evaluates to 8 ...

... • Lambda expressions are earlier methods for describing nameless functions • Lambda expressions are applied to parameter(s) by placing the parameter(s) after the expression e.g., ((x) x * x * x)(2) which evaluates to 8 ...